1. ## Second derivatives

I was searching the board and found a couple posts on this but they didn't really fit the format I was doing derivatives in. $F(x)=-1/(x^2+5)$ is my function. For the first derivative I have $F(x)=2x/(x^2+5)^2$. For some reason, all of my answers I get for the second derivative are wrong. Can anyone show me the steps you take to get to it?

2. $f(x) = -(x^2+5)^{-1}$

$f'(x) = 2x(x^2+5)^{-2}$

$f''(x) = 2x \cdot [-2(x^2+5)^{-3} \cdot 2x] + (x^2+5)^{-2} \cdot 2$

$f''(x) = \frac{-8x^2}{(x^2+5)^3} + \frac{2}{(x^2+5)^2}$

$f''(x) = \frac{-8x^2}{(x^2+5)^3} + \frac{2(x^2+5)}{(x^2+5)^3}$

$f''(x) = \frac{2(5 - 3x^2)}{(x^2+5)^3}$

3. Originally Posted by dougmoser
I was searching the board and found a couple posts on this but they didn't really fit the format I was doing derivatives in. $F(x)=-1/(x^2+5)$ is my function. For the first derivative I have $F(x)=2x/(x^2+5)^2$. For some reason, all of my answers I get for the second derivative are wrong. Can anyone show me the steps you take to get to it?
Set
$F(x):=-\frac{1}{x^{2}+5}.$
Then
$F^{\prime}(x)=\frac{2x}{(x^{2}+5)^{2}}.$
And then
$F^{\prime\prime}(x)=\frac{(2x)^{\prime}(x^{2}+5)^{ 2}-(2x)\big((x^{2}+5)^{2}\big)^{\prime}}{\big((x^{2}+ 5)^{2}\big)^{2}}$ (Chain rule)
......... $=\frac{2(x^{2}+5)^{2}-(2x)\big(2(x^{2}+5)^{\prime}\big)}{(x^{2}+5)^{4}}$ (Chain rule)
......... $=\frac{2(x^{2}+5)^{2}-(2x)\big(2(2x)(x^{2}+5)\big)}{(x^{2}+5)^{4}}$
......... $=-\frac{6x^{4}+20x^{2}-50}{(x^{2}+5)^{4}}$
......... $=-\frac{2(3x^{2}-5)(x^{2}+5)}{(x^{2}+5)^{4}}$
......... $=-\frac{2(3x^{2}-5)}{(x^{2}+5)^{3}}.$

4. Thanks a lot guys!